Chapter 8: Ordinary Differential Equations

Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences.

$\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{}\mathbf{+}\mathbf{}\mathit{y}\mathbf{}\mathbf{=}\mathbf{}{\mathit{e}}^{\mathbf{x}\mathbf{y}}\mathbf{}\mathit{H}\mathit{i}\mathit{n}\mathit{t}\mathbf{:}\mathbf{}\mathit{L}\mathit{e}\mathit{t}\mathbf{}\mathit{u}\mathbf{=}\mathit{x}\mathit{y}$

Evaluate each of the following definite integrals by using the Laplace transform table.

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}\frac{\mathbf{1}}{\mathbf{t}}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{sin}\mathbf{\left(}\mathbf{t}\sqrt{\mathbf{2}}\mathbf{\right)}\mathbf{dt}$

Solve by use of Fourier series. Assume in each case that the right-hand side is a periodic function whose values are stated for one period.

${\mathit{y}}^{\mathbf{"}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{=}\left|x\right|\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{-}\mathbf{\pi }\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{\pi }$.

Solve by use of Fourier series. Assume in each case that the right-hand side is a periodic function whose values are stated for one period.

${\mathit{y}}^{\mathbf{"}}\mathbf{+}\mathbf{9}\mathit{y}\mathbf{=}\{\begin{array}{l}x,0<x<1\\ 0,-1<x<0\end{array}$

Evaluate each of the following definite integrals by using the Laplace transform table.

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}\frac{\mathbf{1}}{\mathbf{t}}{\mathbf{e}}^{\mathbf{-}\mathbf{t}\sqrt{\mathbf{3}}}\mathbf{}\mathbf{sin}\mathbf{2}\mathbf{tcostdt}$

Consider an equation for damped forced vibrations (mechanical or electrical) in which the right-hand side is a sum of several forces or emfs of different frequencies. For example, in (6.32) let the right-hand side be ${\mathit{F}}_{\mathbf{1}}{\mathit{e}}^{\mathbf{i}{\mathbf{\omega }}_{\mathbf{1}}\mathbf{t}}\mathbf{+}{\mathit{F}}_{\mathbf{2}}{\mathit{e}}^{\mathbf{i}{\mathbf{\omega }}_{\mathbf{2}}\mathbf{t}}\mathbf{+}{\mathit{F}}_{\mathbf{3}}{\mathit{e}}^{\mathbf{i}{\mathbf{\omega }}_{\mathbf{3}}\mathbf{t}}$,

Write the solution by the principle of superposition. Suppose, for giventhat we adjust the system so that $\mathit{\omega }\mathbf{=}{\mathit{\omega }}_{\mathbf{1}}^{\mathbf{\text{'}}}$; show that the principal term in the solution is then the first one. Thus, the system acts as a "filter" to select vibrations of one frequency from a given set (for example, a radio tuned to one station selects principally the vibrations of the frequency of that station).

Solve Laplace transforms and the convolution integral or by Green functions.

$\mathbf{y}\mathbf{\text{'}\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}{\mathbf{sec}}^{\mathbf{2}}\mathbf{}\mathbf{t}$

Use the methods to solve the following differential equations. Compare computer solutions and reconcile differences.

$\mathit{y}\mathit{d}\mathit{y}\mathbf{=}\left(-x+\sqrt{x^2+y^2}\right)\mathit{d}\mathit{x}$

Use the methods to solve the following differential equations. Compare computer solutions and reconcile differences.

$\mathit{x}\mathit{y}\mathit{d}\mathit{x}\mathbf{+}\left(y^2-x^2\right)\mathit{d}\mathit{y}\mathbf{=}\mathbf{0}$

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

$\mathit{y}\mathbf{"}\mathbf{+}\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{t}\mathbf{}\mathbf{}\mathbf{}\mathbf{,}$